Indefinite Integration of Rational Functions

Indefinite Integration of Rational Functions

How to solve the indefinite integration of rational functions problems: examples and their solutions.

Example 1

Find the given indefinite integral. The integral of [(x^2 - 3x + 5) / (x - 1)] dx

To integrate a fraction,
change the fraction to make (constant)/(linear) form.

If (numerator's order) ≥ (denominator's order),
then use long division or synthetic division
to make (constant)/(linear) form.

Dividing polynomials (Long division)

Synthetic division

So ∫ (x2 - 3x + 5)/(x - 1) dx = ∫ ( x - 2 + 3/[x - 1] ) dx.

Integrate the given integral.

Then (given) = (1/2)x2 - 2x + 3 ln |x - 1| + C.

Linear change of variable rule

Indefinite integration of 1/x

Example 2

Find the given indefinite integral. The integral of [(3x - 2) / (x(x - 1))] dx

See [3x - 2] / [x(x - 1)].

(numerator's order) < (denominator's order)

So you don't need to use
long division or synthetic division.

But the denominator is not in linear form.

So use partial fraction decomposition
to make (constant)/(linear) form.

Set [3x - 2] / [x(x - 1)] = A/x + B/(x - 1).

Then compare the numerators on both sides
to find A and B.

A = 2, B = 1.

So [3x - 2] / [x(x - 1)] = 2/x + 1/(x - 1).

So (given) = ∫ [2/x + 1/(x - 1)] dx

Integrate the given integral.

Then (given) = 2 ln |x| + ln |x - 1| + C.

Linear change of variable rule

Indefinite integration of 1/x